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Cayley diagram

Q8×D27 #88402

order432
automorphism?
homologyAbelianization (H1) = C23 · Schur multiplier (H2) = C22

Aliases

direct productQ8×D27

Splittings

FittingQ8×C27 ⋊ C2
SylowC27 ⋊ C2×Q8

Subgroups

subgroupG/H
centerC2C22×D27
commutator, cosocleC2×C27C23
FrattiniC2×C9C22×D3
FittingQ8×C27C2
socleC2×C3C22×D9
prime radicalC2×D27C22
hypercenterQ8D27
hypocenterC27C2×Q8
LCSG ▷ C2×C27 ▷ C27
UCSC1 ◁ C2 ◁ Q8

76 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degreeμ(G) = 35
transitive degreeμt(G) = 216

A minimal faithful permutation representation — 3 generators on 35 points:

(2,5)(3,9)(4,12)(6,7)(8,18)(10,11)(13,20)(14,19)(15,25)(16,24)(17,23)(21,22)(26,27);(1,25,24,10,8,5,22,20,23,4,7,26,9,19,14,3,27,6,12,17,13,21,2,18,11,16,15)(28,30,31,35)(29,32,33,34);(28,29,31,33)(30,34,35,32)

Conjugacy classes & character table

75 conjugacy classes — view the full character table.

Relationships

maximal subgroup ofSD16×D27, #11720164, Q16×D27, …
maximal normal subgroup ofSD16×D27, #11720164, Q16×D27, …
maximal quotient of#11720136, #11720137, #11720138, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(432, 49);